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Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (zonostrophic instability), which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg–Landau (G–L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G–L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide range of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G–L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G–L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G–L dynamics that is able to capture the evolution of weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability. [ABSTRACT FROM AUTHOR]

Zonal jets and nonzonal large-scale flows are often present in forced-dissipative barotropic turbulence on a beta plane. The dynamics underlying the formation of both zonal and nonzonal coherent structures is investigated in this work within the statistical framework of stochastic structural stability theory (S3T). Previous S3T studies have shown that the homogeneous turbulent state undergoes a bifurcation at a critical parameter and becomes inhomogeneous with the emergence of zonal and/or large-scale nonzonal flows and that these statistical predictions of S3T are reflected in direct numerical simulations. In this paper, the dynamics underlying the S3T statistical instability of the homogeneous state as a function of parameters is studied. It is shown that, for weak planetary vorticity gradient β, both zonal jets and nonzonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging infinitesimal flow. For large β, the dynamics of the S3T instability differs for zonal and nonzonal flows but in both the destabilizing vorticity fluxes decrease with increasing β. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while nonzonal large-scale flows emerge from resonant and near-resonant triad interactions between the large-scale flow and the stochastically forced eddies. The relation between the formation of large-scale structure through modulational instability and the S3T instability of the homogeneous state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results. [ABSTRACT FROM AUTHOR]

Zonal jets are commonly observed to spontaneously emerge in a β-plane channel from a background of turbulence that is sustained in a statistical steady state by homogeneous stochastic excitation and dissipation of vorticity. The mechanism for jet formation is examined in this work within the statistical wave--mean flow interaction framework of stochastic structural stability theory (SSST) that makes predictions for the emergence of zonal jets in β-plane turbulence. Using the coupled dynamical SSST system that governs the joint evolution of the second-order statistics and the mean flow, the structural stability of the spatially homogeneous statistical equilibrium with no mean zonal jets is studied. It is shown that close to the structural stability boundary, the eddy--mean flow dynamics can be split into two competing processes. The first, which is shearing of the eddies by the local shear described by Orr dynamics in a β plane, is shown in the limit of infinitesimal shear to lead to the formation of jets. The second, which is momentum flux divergence resulting from lateral wave propagation on the nonuniform local mean vorticity gradient, is shown to oppose jet formation. The upgradient momentum fluxes due to shearing of the eddies are shown to act exactly as negative viscosity for an anisotropic forcing and as negative hyperviscosity for isotropic forcing. The downgradient fluxes due to wave flux divergence are shown to act hyperdiffusively. [ABSTRACT FROM AUTHOR]

Large-scale mean flows often emerge in turbulent fluids. In this work, we formulate a stability theory, the stochastic structural stability theory (SSST), for the emergence of jets under external random excitation. We analytically investigate the structural stability of a two-dimensional homogeneous fluid enclosed in a channel and subjected to homogeneous random forcing. We show that two generic competing mechanisms control the instability that gives rise to the emergence of an infinitesimal jet: advection of the eddy vorticity by the mean flow that is shown to be jet forming and advection of the vorticity gradient of the jet by the eddies that is shown to hinder the formation of the mean flow. We show that stochastic forcing with small streamwise coherence and an amplitude larger than a certain threshold leads to the emergence of jets in the channel through a bifurcation of the non-linear SSST system. [ABSTRACT FROM PUBLISHER]

Non-modal mechanisms underlying transient growth of propagating acoustic waves and non-propagating vorticity perturbations in an unbounded compressible shear flow are investigated, making use of closed form solutions. Propagating acoustic waves amplify mainly due to two mechanisms: growth due to advection of streamwise velocity that is typically termed as the lift-up mechanism leading for large Mach numbers to an almost linear increase in streamwise velocity with time and growth due to the downgradient irrotational component of the Reynolds stress leading to linear growth of acoustic wave energy for large times. Synergy between these mechanisms along with the downgradient solenoidal component of the Reynolds stress produces large and robust energy amplification. On the other hand, non-propagating vorticity perturbations amplify due to kinematic deformation of vorticity by the mean flow. For weakly compressible flows, an initial vorticity perturbation abruptly excites acoustic waves with exponentially small amplitude, and the energy gained by vorticity perturbations is transferred back to the mean flow. For moderate Mach numbers, a strong coupling between vorticity perturbations and acoustic waves is found with the energy gained by vorticity perturbations being transferred to acoustic waves that are abruptly excited by the vortex. Calculation of the optimal perturbations for a viscous flow shows that for low Mach numbers, acoustic wave excitation by vorticity perturbations and the subsequent growth of acoustic waves leads to robust energy growth of the order of Reynolds number, while for large Mach numbers, synergy between the lift-up mechanism and the downgradient solenoidal component of the Reynolds stress dominates the growth and leads to a comparable large amplification of streamwise velocity. [ABSTRACT FROM AUTHOR]

Interaction of internal gravity waves with a horizontal shear flow in the absence of potential vorticity perturbations is investigated making use of closed-form solutions. Localized wave packet trajectories are obtained, the energy growth mechanisms occurring are identified, and the potential role of perturbation growth in wave breaking is assessed. Regarding meridional propagation, the wave packet motion is limited by turning levels where the waves are reflected and trapping levels where the waves stagnate. Regarding perturbation energy amplification, two growth mechanisms can be distinguished: growth due to advection of zonal velocity and growth due to downgradient Reynolds stresses. The three-dimensional perturbations producing optimal energy growth reveal that these two mechanisms produce large and robust amplification of zonal velocity and/or density and vertical velocity, potentially leading to shear or convective instability. For large static stability, amplification of density perturbations in conjunction with vertical orientation of the constant phase lines close to the trapping level potentially leads to a convective collapse of the wave packet near the trapping level, in agreement with existing direct numerical simulation studies. For lower static stability and for waves with phase lines oriented horizontally, growth due to advection of zonal velocity dominates, leading to rapid growth of streamwise streaks within the localized wave packet and potentially to shear instability. [ABSTRACT FROM AUTHOR]

Interaction among potential vorticity perturbations and propagating internal gravity waves in a horizontally sheared zonal flow is investigated. In the strong stratification limit, an initial vorticity perturbation weakly excites two propagating gravity waves while the density component of the potential vorticity perturbation is significantly amplified, potentially leading to convective collapse. If stratification is sufficiently weak, a strong coupling between vorticity perturbations and gravity waves is found and spontaneous gravity wave generation occurs. This coupling can be traced to the nonnormal interaction between the potential vorticity and gravity wave manifolds in the weak stratification limit. Vorticity perturbations amplify in energy due to downgradient Reynolds stress when their phase lines tilt against the shear and the large growth attained is transferred to propagating gravity waves. When the flow geometry is such that the excited gravity waves are confined in the vicinity of the vorticity perturbation by their trapping levels, an overall convective collapse of this region can be anticipated. On the other hand, when the flow geometry permits wave propagation, significant gravity wave emission occurs. [ABSTRACT FROM AUTHOR]

Shear flows with a free surface possess diverse branches of modal instabilities. By approximating the mean flow with a piecewise linear profile, an understanding and classification of the instabilities can be achieved by studying the interaction of the edge waves that arise at the density discontinuity at the surface and the vorticity waves that are supported at the mean vorticity gradient discontinuities in the interior. The various branches of instability are identified and their physical origin is clarified. The edge waves giving rise to the modal instabilities can also lead to a modest transient growth that extends into the regions of neutrality of the flow. However, when the continuous spectrum is excited substantial transient growth can arise and the optimal perturbations attain greater energy when compared with the energy of the fastest modal growing perturbation. These optimal perturbations utilize the continuous spectrum to excite at large amplitude the neutral or amplifying modes of the system. [ABSTRACT FROM AUTHOR]

Three-dimensional perturbations producing optimal energy growth in stratified, unbounded constant shear flow are determined. The optimal perturbations are intrinsically three-dimensional in structure. Streamwise rolls emerge as the optimally growing perturbations at long times, but their energy growth factor is limited by stratification to E = O (1/Ri), where Ri is the Richardson number. The perturbations that attain the greatest energy growth in the flow are combinations of Orr solutions and roll solutions that maximize their energy growth in typically O (10) advective time units. These optimal perturbations are localized in the high-shear regions of the boundary layer and are associated with strong updrafts and downdrafts that evolve into streamwise velocity streaky structures in the form of hairpin vortices in agreement with observations. [ABSTRACT FROM AUTHOR]